[PR #10933/597161d backport][3.11] Fix flakey client functional keep alive tests

[bdraco]

(cherry picked from commit 597161d)

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 98.08%. Comparing base (24b0fdd) to head (c16ffe2).
Report is 1 commits behind head on 3.11.

✅ All tests successful. No failed tests found.

Additional details and impacted files

@@           Coverage Diff           @@
##             3.11   #10936   +/-   ##
=======================================
  Coverage   98.08%   98.08%           
=======================================
  Files         126      126           
  Lines       37728    37728           
  Branches     4566     4566           
=======================================
  Hits        37006    37006           
  Misses        544      544           
  Partials      178      178           

Flag	Coverage Δ
CI-GHA	97.97% <100.00%> (ø)
OS-Linux	97.67% <100.00%> (ø)
OS-Windows	94.68% <100.00%> (-0.01%)	⬇️
OS-macOS	96.80% <100.00%> (ø)
Py-3.10.11	96.67% <100.00%> (+<0.01%)	⬆️
Py-3.10.17	97.23% <100.00%> (+0.04%)	⬆️
Py-3.11.12	97.25% <57.14%> (ø)
Py-3.11.9	96.74% <57.14%> (-0.03%)	⬇️
Py-3.12.10	97.70% <100.00%> (-0.05%)	⬇️
Py-3.13.3	97.73% <100.00%> (ø)
Py-3.9.13	96.58% <100.00%> (ø)
Py-3.9.22	97.09% <100.00%> (-0.03%)	⬇️
Py-pypy7.3.16	96.65% <100.00%> (-0.01%)	⬇️
VM-macos	96.80% <100.00%> (ø)
VM-ubuntu	97.67% <100.00%> (ø)
VM-windows	94.68% <100.00%> (-0.01%)	⬇️

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[codspeed-hq]

CodSpeed Performance Report

Merging #10936 will not alter performance

_{Comparing patchback/backports/3.11/597161d1127d04e7842642d7548d4bd8de3115a6/pr-10933 (c16ffe2) with 3.11 (24b0fdd)}

Summary

✅ 55 untouched benchmarks